I. Episodic Volcanism of Tidally Heated Satellites with an Application to Io. II. Thermal State of an Ice Shell on Europa. III. Polar Wander of a Synchronously Rotating Satellite with Application to Europa

Citation

Ojakangas, Gregory Wayne (1988) I. Episodic Volcanism of Tidally Heated Satellites with an Application to Io. II. Thermal State of an Ice Shell on Europa. III. Polar Wander of a Synchronously Rotating Satellite with Application to Europa. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/RMD6-XM66. https://resolver.caltech.edu/CaltechTHESIS:10162009-112521312

Abstract

Two examples of planetary bodies that may have coupled thermal and dynamical evolutions are investigated. The work is presented in three individual papers.

The first example is that of a tidally heated satellite in an orbital resonance, for which the tidal dissipation rate is a strongly increasing function of the internal temperature. For such a satellite, a feedback mechanism exists between the orbital and thermal energies, which may lead to periodic variations in tidal heating within the satellite and its orbital eccentricity. A simple model of this mechanism is presented in the first paper and is applied specifically to Io.

The second example is that of an ice shell on Europa, which is decoupled from the silicate core by a layer of liquid water. In the second paper, the spatially varying thickness that such a shell would have in thermal equilibrium with tidal dissipation within it, surface solar insolation and heat flow from the core is calculated for reasonable rheological laws for ice. The contribution of these variations in ice thickness to Europa's inertia tensor is estimated, and the implications for nonsynchronous rotation of Europa are discussed. In the third paper, a detailed dynamical model is developed, which demonstrates that such a shell may exhibit large—scale polar wander as it approaches thermal equilibrium, because of the destabilizing effect of the variations in ice thickness on the inertia tensor of the shell.

The abstracts of the three papers are reproduced below.

Episodic Volcanism of Tidally Heated Satellites with Application to Io

A simple model of the coupled thermal and orbital evolution of a tidally heated satellite in an orbital resonance is presented and applied specifically to Io. The model quantitatively demonstrates how a feedback mechanism between the orbital and thermal energy of such a satellite can lead to periodic variations in surface heatflow and orbital eccentricity. The convective heatflow and (k/Q) of the satellite are parameterized as local power laws of the temperature, where Q is the quality factor and k is the second-degree tidal potential Love number. The time evolution of the model is determined by two nonlinear equations: an equation governing the orbital eccentricity, and a simple heat-balance equation determining the temperature. A linear stability analysis reveals that the time-independent solution is unstable if n > m + p, where n and m are the exponents in the power laws for (k/Q) and convective heatflow, respectively, and p is the ratio of the convective cooling time scale to the time scale for equilibration of the eccentricity. Numerical integration of the nonlinear equations reveals behavior in qualitative agreement with this relation. Laboratory data on near-solidus peridotites suggest 20 ≾ n ≾ 30, and parameterized convection schemes suggest m ~ 10. Since p is of order unity, it follows that tidally heated satellites are probably in the unstable regime if they are operating near the solidus. It is thus probable that Io has no thermal steady state. The model is made more realistic by (1) arresting the reduction of (k/Q) at low temperature, and (2) arresting the growth of temperature at the mantle solidus and allowing volcanism to remove the excess heat. When the second modification is included, the unstable regime becomes periodic. In addition, a global k substantially larger than the elastic value is possible for a mostly solid Io because the body may begin to behave viscously when the tidal period is longer than, or comparable to, the Maxwell time. This requires a solid-state viscosity of ≾ 4 x 10¹⁵ Pa s, which may be achievable with a small amount of partial melt. The model can easily be adjusted to pass through Io's current observed heatflow (1-2 W m⁻²) and eccentricity (~ 0.004) for reasonable choices of parameters (Q/k)_min ~ 100, (Q/k)_max ~ few x 10³ solidus viscosity ~ 10¹⁵–10¹⁷ Pas, and Q_J within the required dynamical bounds. The periods of high heatflow and acceptable eccentricity typically have duration of ~ 20-20 myr, separated in time by ~ 80-100 myr. Spatial heterogeneities in Io's thermal structure are likely to make the behavior more complicated. The model predicts that Io's mean motion may be currently increasing, a possibility suggested by recent estimates of ṅ₁ from eclipse data. Since Europa's eccentricity mimics that of Io, the model also implies that the tidal stresses in Europa's ice shell may have recently been large enough to produce the observed fracturing. The episodic heating mechanism may be responsible for the resurfacing of Enceladus < 10⁹ years ago.

Thermal State of an Ice Shell on Europa

We consider a model of Europa consisting of an ice shell that is decoupled from a silicate core by a layer of liquid water. The thickness of the shell is calculated as a function of colatitude and longitude, assuming that a state of conductive equilibrium exists with the incident annual average solar insolation, tidal dissipation within the shell, and heat flow from the core. Ice thickness profiles are calculated for each of two plausible rheological behaviors for ice: the Maxwell rheology and the generalized flow law rheology. In both cases the strong temperature—dependence of the dissipation rate is explicitly included as well as the temperature—dependence of the thermal conductivity of ice. Because of the strong temperature dependence of the dissipation rate, nearly all of the tidal dissipation is concentrated in the lowermost few kilometers of the shell. Even though the effective Q of the greater part of the shell is >> 100 in our models, average shell thicknesses do not exceed 25 km. Thus, if the total thickness of H₂O which mantles Europa is ≳ 25 km, none of the models admit the possibility of a completely frozen H₂O layer. The total dissipation rates in our models are comparable to those of a constant Q model with Q ~ 10. The thickness profiles are relatively insensitive to heat flow from the core. The second degree spherical harmonic components of the ice thickness are given and the resulting contributions to the quantities (B-A)/C and (B-C)/A of Europa are estimated. Although the contribution to (B-A)/C is perhaps larger than the permanent value needed to prevent nonsynchronous rotation, its dependence on the shell's orientation relative to synchroneity suggests that very slow nonsynchronous rotation will persist, with reorientation of the shell relative to the satellite-planet direction occurring on a timescale ≳ the thermal diffusion timescale for the shell (~10⁷ yr). The existence of a significant "fossil" bulge on the shell due to long-term elastic behavior of its outer, coldest regions would eliminate nonsynchronous rotation. Since the contribution to ((B-C)/A) of the thickness variations in most of our models is > 0, Europa may experience polar wander as thermal equilibrium is approached, if the above is the most important permanent contribution to ((B-C)/A). The magnitudes of the principal moment differences are insensitive to the details of the parameterization of the tidal dissipation.

Polar Wander of a Synchronously Rotating Satellite with Application to Europa

An ice shell on Europa that is decoupled from the silicate core by a layer of liquid water has a thermal-equilibrium thickness profile that varies with position over its surface, because of spatial variations in the surface temperature and tidal dissipation within the ice (see previous paper). The second spherical harmonic degree components of these thickness variations and of any fossil rotational and tidal bulges present on the shell contribute to the inertia tensor of the body. The problem is that of a planetary elastic lithosphere that is topographically loaded from below. Following the development of Willemann and Turcotte (1981) we develop equations describing the variations in the inertia tensor of a body, which are caused by the addition of second harmonic degree topography to the base of the crust. Applied to the case of an ice shell on Europa, it is found for many choices of parameters that a state of thermal equilibrium for the shell will involve an orientation of Europa's principal axes of inertia (when the hydrostatic bulges are relaxed), which is unusual for a synchronously rotating satellite. Specifically, the intermediate and maximum principal moments are reversed. To reach the preferred orientation for synchronous satellites, a thermal equilibrium ice shell must execute a net reorientation of ninety degrees about the satellite—planet direction. We present a simple model of a rigid, synchronously rotating satellite in a circular orbit for which the difference between the intermediate and maximum principal moments is linear in time, passing through zero when t = 0. The model demonstrates that the expected reorientation is indeed dynamically favored.

We then consider a more realistic model, including the effects of various torques which act to couple the motions of the core, shell, and liquid water layer, as well as the effect of viscous dissipation which arises in the shell due to the predicted polar wander. It is found that the Poincaré torque, gravitational coupling, and the torque due to viscous shear in the liquid water layer are unable to induce significant motion of the core during polar wander of the shell. However, the Poincaré torque exerted on the liquid water layer by the shell is believed to cause the liquid water layer to reorient in coincidence with the shell. The model suggests that viscous friction in the shell eliminates the possibility that polar wander will occur unless preexisting fractures (e.g., due to tidal stresses (Crawford and Stevenson, 1988)) extend from the surface to a depth where the ice behaves viscously on the polar wander time scale. If the temperature T_f at the base of the fractured region is as high as ~ 140-145 K, the model indicates that polar wander occurs on a time scale of 10⁶-10⁵ y (shorter as T_f increases) after the sign of the difference between the maximum and intermediate principal moments reverses. In the absence of dissipation, polar wander would occur in ~ few x 10³ y. Polar wander must occur on a time scale significantly shorter than ~ 10⁷ y, or the thickness profile of the ice will be in continuous equilibrium with its thermal environment regardless of its orientation, and the mechanism driving the polar wander will be virtually eliminated. It is likely that events of large scale polar wander occur episodically, separated in time by periods on the order of the time scale for thermal diffusion through the shell (~ 10⁷ y), although a state of slow, continuous drifting of the pole is also possible. The time scale of viscous flow of topography at the base of the ice is also near 10⁷ y. If dissipation in the shell due to polar wander is a few orders of magnitude smaller than our simple model suggests, polar wander as described here is a much more effective means for fracturing the ice than is tidal flexing, and it may contribute to producing the observed global fracture systems in Europa's ice.

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